1 / 15

Traveling Salesman Problem (TSP)

Traveling Salesman Problem (TSP). Chris Seto Andrea Smith. Problem description. Given a list of a cities and distance between each pair of cities, what is the shortest possible route that visits each? NP-hard problem Problem usually modeled as an undirected graph

ronat
Download Presentation

Traveling Salesman Problem (TSP)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Traveling Salesman Problem (TSP) Chris Seto Andrea Smith

  2. Problem description • Given a list of a cities and distance between each pair of cities, what is the shortest possible route that visits each? • NP-hard problem • Problem usually modeled as an undirected graph • Produces a Hamiltonian Cycle

  3. History • Studies started in the 1800s by Sir William Hamilton and Thomas Kirkman of related problems • Icosian game invented in 1857 • TSP first studied in 1930s by Karl Menger, Hassler Whitney, and Merrill Flood • Solutions appeared in papers in mid-1950s • Determined to be NP-hard in 1972 by Richard M. Karp

  4. Application: Pick And Place • Printed circuit board (PCB) with locations where chips must be placed by robot • The faster the robot can place all components, the faster the PCB will be assembled • The faster the PCB can be assembled, the more PCBs can be made in the same amount of time

  5. Application: Logistics • Warehouse with many parts in various locations • Order is received for several parts which must be picked by robot or employee • What path should the picker follow to ensure that they fill the order in smallest amount of time? • Any number of TSP solutions can be applied and compared in parallel

  6. Application: UPS ORION • Route optimization used by UPS • Implementation started in 2008 • Saves ~35 million miles per year • Increased projected annual savings

  7. NP Hard Solution Methods • Devise an algorithm for an exact solution, even though it may only work efficiently for a small problem • Devise “Suboptimal” heuristic algorithms to yield good, but inexact solutions. • Find special cases for the problem for which better or exact heuristics are developed.

  8. Possible Approaches • Brute-force Method • Best for small number of nodes • Greedy Algorithm • Simplest algorithm for larger number of nodes • Genetic Algorithm • Generates “close to optimal” solutions

  9. Approach: Brute Force • Best for small number of nodes • Number of Hamiltonian Circuits = (n-1)! • Guaranteed an Optimal Solution • Complexity: ((n-1)!) • Tries all possible permutations and compares cost

  10. Brute Force Demo

  11. Approach: Greedy • Prim’s Algorithm, Kruskal’s Algorithm • Not guaranteed the optimal solution • “Close enough” solution • Prim’s Complexity: O() • Kruskal’s Complexity: O()

  12. Approach: Genetic Algorithm • New approach which uses natural selection to create close to optimal solutions • Hamiltonian cycles are continually “bred” with mutations • Crossover occurs between solutions • Relatively quickly produces a solution which is probably close to optimal

  13. Conclusion • Studies originally began 1800s, again in 1950s • Optimal solution found in ((n-1)!)time • Close enough solution found in O() • Used by… • UPS in ORION system • Pick and place • Many other graph representable systems

  14. Questions?

  15. References • http://en.wikipedia.org/wiki/Travelling_salesman_problem • https://xkcd.com/399/ (Comic) • http://www.forbes.com/sites/alexkonrad/2013/11/01/meet-orion-software-that-will-save-ups-millions-by-improving-drivers-routes/ • http://www.theprojectspot.com/tutorial-post/applying-a-genetic-algorithm-to-the-travelling-salesman-problem/5

More Related